Based on the Pauli exclusion principle , no two particles can have the same quantum state. However, in the double slit experiment with electrons (in which we observe wave-particle duality), at some points the wave functions add up to each other. In those specific spatial spots, two electrons have exactly the same quantum states, but not only do they not exclude each other, they are adding to the probability of each other's presence. How that is explained? Is this because only the two electrons with different spins are adding up to each other in this experiment? If so, I think the number of electrons in those specific points should be statistically half of the expected. Is that true?
The text of your question implies that you have misunderstood the theoretical description of the Young's slits experiment. In the interference here, it is not two different electrons whose wavefunctions interfere, but two parts of the wavefuntion of each individual electron. Of course in practice many electrons may pass through the system, but each interferes with itself.
If it happened that more than two electrons approached the slits at the same time, then the Pauli exclusion principle would prevent them all having the same spatial wavefunction. That doesn't mean some would pass through one slit, some through another however. It means that the joint wavefunction would involve a sum of terms with each electron at both slits, but with various entanglements and minus signs between parts with electron labels swapped (I won't go into the details). However the more simple concept of Coulomb repulsion would already cause a big effect in this case.
If we were working with neutrons then there would be no Coulomb repulsion. With bright enough beams, Pauli exclusion would be relevant. There would still be interference etc. but the theoretical description would be more lengthy. One would have to write down a joint state for multiple neutrons; lots of entanglement as in previous paragraph.
The Pauli exclusion principle is an empirical result.
It's theoretically justified by positing fermionic and bosonic wave functions which are, respectively anti-symmetric and symmetric and which, respectively, gain a sign change or are unchanged when two particles are swapped, in the first case, two fermions, and in the latter, two bosons.
The phenomena of particle wave duality does not distinguish between the two cases, and applies to both; in fact, recall that particle-wave duality was first discovered with photons, and these are the gauge bosons of the Electromagnetic force, and it was the inspired idea of de Broglie that posited the same effect may occur with electrons, which are fermions, and when this idea was experimentally put to the tested, it was found to be the case.
Hence your headline question is somewhat ill-concieved.
The Pauli Exclusion Principle applies to potential wells in the subatomic realm, like the shells of the electron cloud around atomic nuclei. In Young's dual slit experiment the scale is much larger than the subatomic realm. Small groups of electrons are fired down a cathode ray tube, through the dual slit grating and impact a screen covered in a material that phosphoresces when struck by electrons. The pattern that appears is the record of where the electrons struck. Not the electrons themselves. Feynman did provide a thought experiment where a singe electron was used in Young's dual slit experiment. Here is an article describing a realization of Feynman's thought experiment.
But again the interference pattern is the record of the location of the electrons, not the electrons themselves. The screen is not a single potential well where all the electrons are piling up as they pass through the dual slit grating. So as @Luke Pritchett pointed out PEP does not apply.